I have seen that still there some theme having bundled with old files for the add to cart template being used for variable products, adding a variable product to cart fails. This should be solved by the theme developers by updating the template file.
In order to solve this woocommerce upgrade issue, the quick solution is to delete the below mentioned file from the theme directory.
This will make WooCommerce use the new bundled template file and your variable products will add to cart again. Once your theme developer updates your theme, please check if it actually works as that will overwrite this change again.